Kneza Mihaila 35, 11000 Belgrade, Serbia and Montenegro ..... now have a C++ library that defines the new Word data type and implements all the operators that the hash ..... In Proceedings of the 17th National Conference on AI and 12th.

Sep 9, 2002 - I would also like to thank my family for their love and support. ..... In the case of collision detection between cloth and skin we are dealing with ..... space and contains a hyperplane that partitions that region into front and back.

Data Integrity • Cryptographic Hash Function: Provides assurance of data integrity • Let h be a hash function and x some data. • The hash creates a fingerprint of the data, often referred to as the message digest. • Typically, x is a large binary string • The digest is a fairly short binary string, say 160 bits.

Applications • Say y=h(x), and y is stored in some secured place. • If x is altered to say x’ and if we assume that h(x)≠h(x’), then the alteration of the message is readily caught, by verifying y≠y’, where y’=h(x’) • Used in digital signature schemes • Used for message authentication codes (MAC)

Low Power Ajit Pal IIT Kharagpur

2

Application: Data Integrity

Application: Digital Signatures

Low Power Ajit Pal IIT Kharagpur

3

A Keyed Hash Function • Suppose we also have a key in the computation of the hash functions. • y=hK(x), and the key is kept secret. – – – – –

Alice and Bob share K Alice computes y for x, using K and sends to Bob. Bob receives x’ and computes the hash value. If the hashes match, the message is unaltered. Note that here y is not required to be kept secret. Why?

What is a Cryptographic Hash Family?

• Note: X could be finite or infinite set, but Y is always finite • If |X|=N, |Y|=M, then there are MN possible FX,Y (the cardinality of the set of all functions from X to Y) • Any hash family, F ⊆ F X ,Y is called an (N,M) hash family.

Low Power Ajit Pal IIT Kharagpur

4

Security of Hash Functions • There are three important properties which a hash function must satisfy. • The properties are required for the security of the applciations. – Preimage – Second Preimage – Collision

• We define them one by one.

Preimage

• If the Preimage can be solved then (x,y) is a valid pair. • A hash function for which Preimage cannot be efficiently solved is said to be preimage resistant.

Low Power Ajit Pal IIT Kharagpur

5

Second Preimage

• If this problem is solved, then the pair (x’,h(x)) is valid • If it cannot be done efficiently then the hash is Second Preimage resistant.

Collision

• Note that if this is solved, then if (x,y) is a valid pair so is (x’,y) • If not (efficiently solvable) the hash function is called collision resitant

Low Power Ajit Pal IIT Kharagpur

6

The Random Oracle Model • Captures the concept of an ideal hash function • If a hash function, h is ideal then the only way to compute the hash of a given value is by actually computing it: i,e even if many previous values are known.

A Non-Ideal Hash Function • Consider a hash function h: Zn Æ Zn which is a linear function, say – h(x,y)=ax + by mod n, a, b ε Zn, n≥2 is a positive integer – Suppose, h(x1,y1)=ax1+by1, h(x2,y2)=ax2+by2 h(rx1+sx2 mod n, ry1+sy2 mod n)= =rh1(x1,y1)+sh2(x2,y2) mod n Thus we can compute the hash of another value apart from (x1,y1) and (x2,y2) without actually computing the hash value. We are computing the new hash value from pre-computed values Note that we do not require the knowledge of a and b also. This is not what is an ideal hash function according to the RO model.

Low Power Ajit Pal IIT Kharagpur

7

What is an Oracle? • It is not an algorithm • neither a formula • imagine this to be a giant book of random numbers and each page is a value x and the number written on that page is h(x)

An Independence Theorem

• Note that the above is a conditional probability • It states that the knowledge of the previously computed values, does not give any advantage to the future computations of h(x) • This assumption in the RO model will be used in the complexity proofs that follow.

Low Power Ajit Pal IIT Kharagpur

8

Algorithms in the RO model • These algorithms are applicable to all hash functions, since the algorithms are not dependent on the details of the hashing method. • These algorithms are randomized, in the sense that they make random choices • In particular they can fail, but if they succeed they are correct: Las Vegas Algorithms

Algorithms in the RO model • Worst case success probability, ε: if for every problem instance, the randomized algorithm returns a correct answer with probability at least ε • Average case success probability: if the probability that the algorithm returns a correct answer, averaged over all problem instances, is at least ε • The average success probability is averaged over all possible random choices of FX,Y, and all possible random choices of xεX and/or yεY, if x and/or y are specified as a part of the problem instance.

Low Power Ajit Pal IIT Kharagpur

9

Algorithm Find-Preimage

:

Algorithm Find-Second Preimage

Low Power Ajit Pal IIT Kharagpur

10

Algorithm FindCollision

Relating Q and ε

• So, if we hash little over sqrt(M) values, we have a 50% chance of collision • Thus our algorithm is (1/2, O(sqrt(M)) algorithm

Low Power Ajit Pal IIT Kharagpur

11

Comparison of Security Criteria • Solving Collision is easier than solving Preimage or 2nd Preimage • Can we reduce one problem to the other? • We shall study two reductions: – Collision to 2nd Preimage – Collision to Preimage

Proof Method • Reducing Collision to Preimage: – Assume that Preimage can be solved using a randomized algorithm – Show that then the Collision can be solved.

• CollisionHardness Preimage Resistance

Low Power Ajit Pal IIT Kharagpur

12

The first reduction

• Oracle-2nd-Preimage is an (ε,q) algorithm. • Since it is a Las-Vegas algorithm, if it gives an answer it will be correct. Thus, x≠x’ and h(x)=h(x’). Thus the collision is also found. • Thus Collision-to-second-preimage is also an (ε,q) Las-Vegas algorithm

The second reduction

• Assume that Oracle-Preimage is a (1,Q) Las Vegas algorithm • We will make some weak assumptions on the size of X and Y, |X|≥2|Y|

Low Power Ajit Pal IIT Kharagpur

13

Reduction

• Proof discussed in class.

Point to Ponder • If the OraclePreimage has a success probability of ε